Question: Camille is attending a fundraiser. She pays for her admission and buys raffle tickets for $\$5$ each. If she buys $10$ raffle tickets, then she would spend a total of $\$135$ at the fundraiser. The number $S$ of dollars Camille spends at the fundraiser is a function of $r$, the number of raffle tickets she buys. Write the function's formula. $S=$
Answer: The price of each raffle ticket is constant, so we're dealing with a linear relationship. We could write the desired formula in slope-intercept form: $S= mr+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that the amount of money Camille spends increases by $\$5$ for each raffle ticket she buys, so the slope $ m$ is ${5}$, and our function looks like $S={5}r+ b$. We also know that if Camille buys $10$ raffle tickets, then she would spend a total of $\$135$ at the fundraiser, which means that when $r=10$, $S=135$. We can substitute this into the formula of the function to find $ b$ : $\begin{aligned}{5}\cdot10+ b&=135\\\\ 50+ b&=135\\\\ b&={85}\end{aligned}$ This means admission to the fundraiser costs $\$85$. Since $ m = {5}$ and $ b = {85}$, the desired formula is: $S={5}r+{85}$